Finite Group Databasebeta

← C2×#20241506 #26131217

Subgroups — C2×#20241506 #26131217

Order 12032.

1202 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093941C211yesmaximal, characteristic
2C22×#13093939C211yesmaximal, characteristic
3#20241506C214yesmaximal, direct product factor
4C2×C47×#386C211yesmaximal, characteristic
5C22×#10625329—21no—
6C22×C22⋊Dic47—21no—
7C2×#13093941—22no—
8C2×#13093941—21no—
9C2×#13093941C414yescomplemented
10C2×#13093941C2212yes—
11C2×#13093939—21no—
12C2×#13093939—22no—
13C2×#13093939C414yescomplemented
14C2×#13093939C2212yes—
15C23×D4×C47C2211yescharacteristic
16C24×C4×C47C411yescharacteristic, complemented
17C22×C22⋊C4×C47—21no—
18C22×C22⋊C4×C47C411yescharacteristic, complemented
19C47×#386—21no—
20C47×#386—21no—
21C47×#386C412yescomplemented
22C47×#386C2212yes—
23C2×#95×C47—21no—
24C23×Dic47—41no—
25C2×#10625329—24no—
26C2×#10625329—22no—
27C2×#10625329—42no—
28C2×#10625329—41no—
29C22×C47⋊C8—41no—
30C2×C22⋊Dic47—24no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—41no—
33C2×C22⋊Dic47—42no—
34#13093941—24no—
35#13093941—22no—
36#13093941C2×C414yescomplemented
37#13093939—24no—
38#13093939—22no—
39#13093939C2×C414yescomplemented
40C25×C47C2×C411yescharacteristic
41C25×C47D411yescharacteristic
42C22×D4×C47—41no—
43C22×D4×C47—42no—
44C22×D4×C47—41no—
45C22×D4×C47C2×C411yescharacteristic
46C22×D4×C47C2×C412yes—
47C22×D4×C47C2×C412yes—
48C22×D4×C47D411yescharacteristic
49C22×D4×C47D412yes—
50C22×D4×C47D411yescharacteristic
51C22×D4×C47Q812yes—
52C22×D4×C47C2311yescharacteristic
53C23×C4×C47—21no—
54C23×C4×C47—21no—
55C23×C4×C47—21no—
56C23×C4×C47—44no—
57C23×C4×C47—41no—
58C23×C4×C47—41no—
59C23×C4×C47C2×C412yescomplemented
60C23×C4×C47D411yescharacteristic
61C2×C22⋊C4×C47—22no—
62C2×C22⋊C4×C47—21no—
63C2×C22⋊C4×C47—22no—
64C2×C22⋊C4×C47—44no—
65C2×C22⋊C4×C47—42no—
66C2×C22⋊C4×C47C2×C412yescomplemented
67#95×C47—22no—
68#95×C47—22no—
69C24×C47—22no—
70C24×C47—21no—
71C24×C47—21no—
72C24×C47—21no—
73C24×C47—22no—
74C24×C47—21no—
75C24×C47—42no—
76C24×C47—42no—
77C24×C47—42no—
78C24×C47—42no—
79C24×C47C4212yes—
80C24×C47C22⋊C411yescharacteristic
81C24×C47C22⋊C412yes—
82C24×C47C22⋊C411yescharacteristic
83C24×C47C22⋊C411yescharacteristic
84C24×C47C4⋊C412yes—
85C24×C47C4⋊C412yes—
86C24×C47C22×C411yescharacteristic
87C24×C47C2×D411yescharacteristic
88C2×D4×C47—42no—
89C2×D4×C47—42no—
90C2×D4×C47—42no—
91C2×D4×C47—42no—
92C2×D4×C47—41no—
93C2×D4×C47—41no—
94C2×D4×C47—42no—
95C2×D4×C47—42no—
96C2×D4×C47—41no—
97C2×D4×C47—41no—
98C2×D4×C47—41no—
99C2×D4×C47—41no—
100C2×D4×C47—42no—
101C2×D4×C47—42no—
102C2×D4×C47—41no—
103C2×D4×C47—41no—
104C2×D4×C47C4212yes—
105C2×D4×C47C22⋊C412yes—
106C2×D4×C47C22⋊C412yes—
107C2×D4×C47C4⋊C412yes—
108C2×D4×C47C4⋊C412yes—
109C2×D4×C47C4⋊C412yes—
110C2×D4×C47C22×C411yescharacteristic
111C2×D4×C47C22×C411yescharacteristic
112C2×D4×C47C2×D411yescharacteristic
113C2×D4×C47C2×Q811yescharacteristic
114C22×C4×C47—21no—
115C22×C4×C47—22no—
116C22×C4×C47—21no—
117C22×C4×C47—24no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—44no—
121C22×C4×C47—44no—
122C22×C4×C47—42no—
123C22×C4×C47—42no—
124C22×C4×C47—48no—
125C22×C4×C47—44no—
126C22×C4×C47—41no—
127C22×C4×C47—41no—
128C22×C4×C47—42no—
129C22×C4×C47—44no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47C22⋊C411yescharacteristic
134C22×C4×C47C22⋊C412yes—
135C22×C4×C47C4⋊C412yes—
136C22×C4×C47C2×D411yescharacteristic
137C22×Dic47—44no—
138C22×Dic47—42no—
139C22×Dic47—82no—
140C22×Dic47—82no—
141C22⋊C4×C47—44no—
142C22⋊C4×C47—44no—
143C22⋊C4×C47—42no—
144C22⋊C4×C47—42no—
145C22⋊Dic47—24no—
146C22⋊Dic47—44no—
147C22⋊Dic47—42no—
148#10625329—24no—
149#10625329—44no—
150#10625329—42no—
151C2×C47⋊C8—44no—
152C2×C47⋊C8—42no—
153C23×C47—22no—
154C23×C47—22no—
155C23×C47—22no—
156C23×C47—22no—
157C23×C47—22no—
158C23×C47—22no—
159C23×C47—22no—
160C23×C47—22no—
161C23×C47—21no—
162C23×C47—21no—
163C23×C47—21no—
164C23×C47—21no—
165C23×C47—21no—
166C23×C47—21no—
167C23×C47—22no—
168C23×C47—22no—
169C23×C47—21no—
170C23×C47—21no—
171C23×C47—41no—
172C23×C47—41no—
173C23×C47—44no—
174C23×C47—41no—
175C23×C47—44no—
176C23×C47—41no—
177C23×C47—42no—
178C23×C47—42no—
179C23×C47—48no—
180C23×C47—41no—
181C23×C47—41no—
182C23×C47—44no—
183C23×C47—41no—
184C23×C47—44no—
185C23×C47—41no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—81no—
191C23×C47—81no—
192C23×C47—84no—
193C23×C47#9512yes—
194C23×C47#9512yes—
195C23×C47#9512yes—
196C23×C47#9911yescharacteristic
197C23×C47#9912yes—
198C23×C47#9911yescharacteristic
199C23×C47C2×C4211yescharacteristic
200C23×C47C2×C22⋊C411yescharacteristic
201C23×C47C2×C22⋊C411yescharacteristic
202C23×C47C2×C4⋊C411yescharacteristic
203C23×C47C2×C4⋊C411yescharacteristic
204D4×C47—42no—
205D4×C47—42no—
206D4×C47—42no—
207D4×C47—42no—
208D4×C47—42no—
209D4×C47—42no—
210D4×C47—41no—
211D4×C47—41no—
212D4×C47—41no—
213D4×C47—41no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—24no—
223C2×C4×C47—42no—
224C2×C4×C47—44no—
225C2×C4×C47—48no—
226C2×C4×C47—44no—
227C2×C4×C47—44no—
228C2×C4×C47—42no—
229C2×C4×C47—44no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—84no—
234C2×C4×C47#9512yes—
235C2×C4×C47C2×C22⋊C411yescharacteristic
236C2×C4×C47C2×C4⋊C411yescharacteristic
237C2×Dic47—44no—
238C2×Dic47—82no—
239C2×Dic47—82no—
240C2×Dic47—88no—
241C47⋊C8—44no—
242C2×#1390—471nomaximal, Sylow
243C22×C47—21no—
244C22×C47—21no—
245C22×C47—21no—
246C22×C47—21no—
247C22×C47—24no—
248C22×C47—24no—
249C22×C47—21no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—21no—
253C22×C47—21no—
254C22×C47—21no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—41no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47—44no—
264C22×C47—44no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—41no—
272C22×C47—41no—
273C22×C47—44no—
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47—81no—
278C22×C47—81no—
279C22×C47—82no—
280C22×C47—82no—
281C22×C47#34212yes—
282C22×C47#34212yes—
283C22×C47AΣL(1,𝔽16)12yes—
284C22×C47#35212yes—
285C22×C47C2×#9511yescharacteristic
286C22×C47C2×#9911yescharacteristic
287C22×C47C2×#9911yescharacteristic
288C4×C47—21no—
289C4×C47—22no—
290C4×C47—21no—
291C4×C47—44no—
292C4×C47—82no—
293Dic47—84no—
294C22×#100—471no—
295C22×#99—471no—
296C2×#386D4711yescharacteristic
297#1390—474no—
298C2×C47—21no—
299C2×C47—21no—
300C2×C47—22no—
301C2×C47—41no—
302C2×C47—42no—
303C2×C47—41no—
304C2×C47—41no—
305C2×C47—42no—
306C2×C47—41no—
307C2×C47#139014yescomplemented
308C2×C47C2×#34211yescharacteristic
309C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
310C2×C47C2×#35211yescharacteristic
311C23×D4C2×D4711yescharacteristic
312C24×C4Dic4711yescharacteristic, complemented
313C22×M16—941no—
314C22×C22⋊C4—21no—
315C22×C22⋊C4—941no—
316C22×C22⋊C4Dic4711yescharacteristic, complemented
317C2×#100—474no—
318C2×#100—472no—
319C2×#100—942no—
320C2×#100—941no—
321C2×#99—474no—
322C2×#99—472no—
323C2×#99—942no—
324C2×#99—941no—
325#386—21no—
326#386—21no—
327#386Dic4712yescomplemented
328#386C2×D4712yes—
329C2×#95—21no—
330C47C2×#139011yescharacteristic, complemented, Sylow
331C25C2×Dic4711yescharacteristic
332C25C22⋊D4711yescharacteristic
333C22×D4—42no—
334C22×D4—41no—
335C22×D4—41no—
336C22×D4Dic9412yes—
337C22×D4C4×D4712yes—
338C22×D4C4×D4712yes—
339C22×D4D18812yes—
340C22×D4C2×Dic4711yescharacteristic
341C22×D4C22⋊D4711yescharacteristic
342C22×D4C22⋊D4711yescharacteristic
343C22×D4C22×D4711yescharacteristic
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—44no—
348C23×C4—41no—
349C23×C4—41no—
350C23×C4—1881no—
351C23×C4C2×Dic4712yescomplemented
352C23×C4C22⋊D4711yescharacteristic
353C2×M16—944no—
354C2×M16—942no—
355C2×M16—1881no—
356C2×M16—1882no—
357C22×C8—1881no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—44no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—944no—
364C2×C22⋊C4—942no—
365C2×C22⋊C4—1882no—
366C2×C22⋊C4—1881no—
367C2×C22⋊C4C2×Dic4712yescomplemented
368#100—474no—
369#100—944no—
370#100—942no—
371#99—474no—
372#99—944no—
373#99—942no—
374#95—22no—
375#95—22no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24C4×Dic4712yes—
387C24Dic47⋊C412yes—
388C24C4⋊Dic4712yes—
389C24(C2×D47)⋊C412yes—
390C24C22⋊Dic4711yescharacteristic
391C24C22⋊Dic4711yescharacteristic
392C24C22⋊Dic4711yescharacteristic
393C24C22×Dic4711yescharacteristic
394C24C2×C22⋊D4711yescharacteristic
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4C4×Dic4712yes—
412C2×D4Dic47⋊C412yes—
413C2×D4Dic47⋊C412yes—
414C2×D4C4⋊Dic4712yes—
415C2×D4(C2×D47)⋊C412yes—
416C2×D4(C2×D47)⋊C412yes—
417C2×D4C2×Dic9411yescharacteristic
418C2×D4C2×C4×D4711yescharacteristic
419C2×D4C2×C4×D4711yescharacteristic
420C2×D4C2×D18811yescharacteristic
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—3762no—
441C22×C4—3762no—
442C22×C4—1882no—
443C22×C4—1884no—
444C22×C4Dic47⋊C412yes—
445C22×C4(C2×D47)⋊C412yes—
446C22×C4C22⋊Dic4711yescharacteristic
447C22×C4C2×C22⋊D4711yescharacteristic
448M16—944no—
449M16—1882no—
450M16—1884no—
451C2×C8—1882no—
452C2×C8—1884no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—944no—
458C22⋊C4—1882no—
459C22⋊C4—1884no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—44no—
481C23—44no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—48no—
487C23—41no—
488C23—41no—
489C23—44no—
490C23—44no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—81no—
498C23—81no—
499C23—84no—
500C23C47⋊#9512yes—
501C23C47⋊#9512yes—
502C23C47⋊#9512yes—
503C23#1309393911yescharacteristic
504C23#1309393912yes—
505C23#1309393911yescharacteristic
506C23C2×C4×Dic4711yescharacteristic
507C23C2×(C2×D47)⋊C411yescharacteristic
508C23C2×C22⋊Dic4711yescharacteristic
509C23C2×Dic47⋊C411yescharacteristic
510C23C2×C4⋊Dic4711yescharacteristic
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—84no—
541C2×C4—3762no—
542C2×C4—3762no—
543C2×C4—3768no—
544C2×C4—1884no—
545C2×C4C47⋊#9512yes—
546C2×C4C2×(C2×D47)⋊C411yescharacteristic
547C2×C4C2×Dic47⋊C411yescharacteristic
548C8—1884no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—24no—
554C22—24no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22#1568889012yes—
588C22#1568889012yes—
589C22#1568891012yes—
590C22#1568891312yes—
591C22C2×C47⋊#9511yescharacteristic
592C22C2×#1309393911yescharacteristic
593C22C2×#1309393911yescharacteristic
594C4—21no—
595C4—22no—
596C4—21no—
597C4—44no—
598C4—82no—
599C4—3764no—
600C2—21no—
601C2—21no—
602C2—22no—
603C2—41no—
604C2—42no—
605C2—41no—
606C2—41no—
607C2—42no—
608C2—41no—
609C2#2024150614yesdirect product factor
610C2C2×#1568889011yescharacteristic
611C2C2×#1568891011yescharacteristic
612C2C2×#1568891311yescharacteristic
subgroupG/H
centerC23#13093939
commutatorC23×C47C2×C42
FrattiniC22×D4C22×D47
FittingC2×C47×#386C2
socleC23×C47#99
prime radicalC23×D4×C47C22
cosocleC22×D4×C47C23
hypercenterC2×#386D47
hypocenterC47C2×#1390
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ C2×#386

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20241506C2direct (N × H)
9. C2×#13093941C4semidirect
13. C2×#13093939C4semidirect
16. C24×C4×C47C4semidirect
18. C22×C22⋊C4×C47C4semidirect
21. C47×#386C4semidirect
36. #13093941C2×C4semidirect
39. #13093939C2×C4semidirect
59. C23×C4×C47C2×C4semidirect
66. C2×C22⋊C4×C47C2×C4semidirect
307. C2×C47#1390semidirect
312. C24×C4Dic47semidirect
316. C22×C22⋊C4Dic47semidirect
327. #386Dic47semidirect
330. C47C2×#1390semidirect
351. C23×C4C2×Dic47semidirect
367. C2×C22⋊C4C2×Dic47semidirect
609. C2#20241506direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#130939412
C22×#130939392
#202415062
C2×C47×#3862
C2×#139047